Uncertainty Quantification with the Probabilistic Collocation Method

Probabilistic stream flow for a small basin simulation generation using the Probabilistic Collocation Method with uncertain input. The above graphs shows the mean model response and the upper and lower inner quartile ranges for stream flow.

Probabilistic stream flow for a small basin simulation generation using the Probabilistic Collocation Method with uncertain input. The above graphs shows the mean model response and the upper and lower inner quartile ranges for stream flow.

As a result of today's high performance computing resources, sophisticated numerical models are capable of incorporating many complex processes. As is often the case, the partial or incomplete knowledge of these processes, and the input parameters required to describe them, necessitates the modeller to make various assumptions and approximations, and in doing so, introduces uncertainty into the numerical model. The aim of parametric uncertainty quantification is to measure the variability in model responses of interest propagated by uncertain model inputs. In doing so, uncertainty quantification lends credence to the predications made by numerical models. In practice, Monte Carlo or Latin Hypercube simulation are the standard workhorse methods for uncertainty quantification. Although these methods are robust, scale well to higher dimensions, and are trivially parallelizable, they are very often slow to converge, typically requiring many hundreds to thousands of simulations to obtain an acceptable level of accuracy. In integrated hydrologic simulations that consider both surface water and groundwater effects, a single simulation may require on the order of days or weeks to finish, such methods are infeasible.     

To overcome these limitations we are investigating a more sophisticated method for representing parametric uncertainty, namely, non-intrusive generalized polynomial chaos (gPC), which has not previously been applied to fully-integrated surface and subsurface flow and transport models. In contrast to sampling methods, gPC represents uncertain model input parameters as random variables and represents a model response of interest as a weighted sum of orthogonal polynomials over the random inputs. The increased complexity of this approach is offset by its potential for fast convergence (exponential rate) and significant speedup over existing standard approaches. For example, in other fields gPC has been shown to compute the mean and variance to the same accuracy as Monte Carlo or Latin Hypercube with between 10–100 times fewer simulations. The aim of this research is to determine whether gPC is suitable for representing parametric uncertainty in integrated hydrologic simulations and if it can be tailored to deliver efficient performance and accurate results in this application domain.